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Bernoulli's equation relates the energy conservation in a fluid moving along a streamline. The equation applies to incompressible and inviscid fluids under steady flow. For such a flow, Newton's second law is applied to a small fluid element, which experiences forces due to pressure differences, gravity, and velocity variations. The force balance leads to the following form of Bernoulli's equation:
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In a fluid at rest, the pressure at any point beneath the fluid surface depends solely on the depth, not on the container's shape or size. This principle, known as hydrostatic pressure, arises because, in stationary fluids, there is no acceleration, meaning the forces within the fluid balance out. Only vertical forces, caused by the weight of the fluid above, contribute to pressure changes with depth.
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Testing the Markov hypothesis in fluid flows.

Daniel W Meyer1, Frédéric Saggini1

  • 1Institute of Fluid Dynamics, ETH Zürich Sonneggstrasse 3, CH-8092 Zurich, Switzerland.

Physical Review. E
|June 15, 2016
PubMed
Summary

We developed methods to test the Markov hypothesis crucial for modeling turbulent and subsurface flow. Markov models accurately describe particle transport in heterogeneous media above a specific scale.

Area of Science:

  • Physics
  • Environmental Science
  • Applied Mathematics

Background:

  • Stochastic Markov processes are fundamental for modeling complex systems like fluid dynamics and subsurface contaminant transport.
  • The Markov hypothesis, central to these models, assumes future states depend only on the present, not past history.
  • Validating this hypothesis is crucial for the reliability of predictive models in various scientific domains.

Purpose of the Study:

  • To introduce and validate novel methods for rigorously testing the Markov hypothesis in physical systems.
  • To assess the applicability and limitations of Markovian models for particle dynamics in turbulent flows and heterogeneous subsurface environments.
  • To investigate the influence of medium heterogeneity and particle type on the validity of the Markov assumption.

Main Methods:

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  • Utilizing the weak Chapman-Kolmogorov equation and the strong Markov condition to formulate hypothesis testing procedures.
  • Applying the developed methodology to analyze the trajectories of fluid and inertial particles in turbulent flows.
  • Evaluating the Markovian behavior of fluid particles within heterogeneous subsurface media with varying log-conductivity correlation structures.

Main Results:

  • The study demonstrates the effectiveness of the proposed methodology in discerning Markovian from non-Markovian behavior.
  • In subsurface macrodispersion, Markov models were found to be applicable above a certain scale of interest, contingent on heterogeneity levels and log-conductivity correlation.
  • Unexpected similarities were observed in the velocity dynamics across different tested media, including turbulent and subsurface environments.

Conclusions:

  • The developed methods provide a robust framework for validating the Markov hypothesis in complex physical processes.
  • The findings suggest that while the Markov assumption has limitations, it can be a valid approximation for particle transport in certain subsurface conditions and scales.
  • The observed similarities in velocity dynamics warrant further investigation into universal scaling laws governing particle motion across diverse complex media.